Computational Two-Party Correlation: A Dichotomy for Key-Agreement Protocols

TL;DR

This paper establishes a dichotomy for efficient two-party protocols, showing they are either uncorrelated or imply key-agreement, and applies this to problems in cryptography and privacy.

Contribution

It introduces new tools to analyze the correlation structure of two-party protocols and proves a fundamental dichotomy theorem classifying all such protocols.

Findings

Protocols are either uncorrelated or imply key-agreement

Uncorrelated protocols have trivial correlation between outputs

The dichotomy advances understanding of cryptographic assumptions for privacy mechanisms

Abstract

Let $\pi$ be an efficient two-party protocol that given security parameter $\kappa$, both parties output single bits $X_\kappa$ and $Y_\kappa$, respectively. We are interested in how $(X_\kappa,Y_\kappa)$ "appears" to an efficient adversary that only views the transcript $T_\kappa$. We make the following contributions: $\bullet$ We develop new tools to argue about this loose notion and show (modulo some caveats) that for every such protocol $\pi$, there exists an efficient simulator such that the following holds: on input $T_\kappa$, the simulator outputs a pair $(X'_\kappa ,Y'_\kappa)$ such that $(X'_\kappa,Y'_\kappa,T_\kappa)$ is (somewhat) computationally indistinguishable from $(X_\kappa,Y_\kappa,T_\kappa)$. $\bullet$ We use these tools to prove the following dichotomy theorem: every such protocol $\pi$ is: - either uncorrelated -- it is (somewhat) indistinguishable from an efficient protocol whose parties interact to produce $T_\kappa$, but then choose their outputs independently from some product distribution (that is determined in poly-time from $T_\kappa$), - or, the protocol implies a key-agreement protocol (for infinitely many $\kappa$'s). Uncorrelated protocols are uninteresting from a cryptographic viewpoint, as the correlation between outputs is (computationally) trivial. Our dichotomy shows that every protocol is either completely uninteresting or implies key-agreement. $\bullet$ We use the above dichotomy to make progress on open problems on minimal cryptographic assumptions required for differentially private mechanisms for the XOR function. $\bullet$ A subsequent work of Haitner et al. uses the above dichotomy to makes progress on a longstanding open question regarding the complexity of fair two-party coin-flipping protocols.